Geodesic
On the β-Construction in K-Theory
Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 819-824

Voir la notice de l'article provenant de la source Cambridge University Press

The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ. 
Naylor, C. M. On the β-Construction in K-Theory. Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 819-824. doi: 10.4153/CJM-1972-080-1
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[1] 1. Atiyah, M. F., On the K-theory of compact Lie groups, Topology 4 (1965), 95–99. Google Scholar

[2] 2. Bott, R., Lectures on K(X) (Benjamin, New York, 1969). Google Scholar

[3] 3. Bott, R. and Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029. Google Scholar

[4] 4. Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. 6 (1957), 111–244. Google Scholar

[5] 5. Hodgkin, L., On the K-theory of Lie groups, Topology 6 (1967), 1–36. Google Scholar

[6] 6. Samelson, H., Notes on Lie Algebras (Van Nostrand, New York, 1969). Google Scholar

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